Precalculus Examples

$\log_{3} ({(3)}^{x - 6}) - \log_{3} ({(3)}^{x})$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{3} (\frac{{(3)}^{x - 6}}{{(3)}^{x}})$

Step 2

Rewrite $\frac{{(3)}^{x - 6}}{{(3)}^{x}}$ as ${(3)}^{x - 6} {({(3)}^{x})}^{- 1}$ .

$\log_{3} ({(3)}^{x - 6} {({(3)}^{x})}^{- 1})$

Step 3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{3} ({(3)}^{x - 6} \cdot 3^{x \cdot - 1})$

Step 4

Rewrite $\log_{3} ({(3)}^{x - 6} \cdot 3^{x \cdot - 1})$ as $\log_{3} ({(3)}^{x - 6}) + \log_{3} (3^{x \cdot - 1})$ .

$\log_{3} ({(3)}^{x - 6}) + \log_{3} (3^{x \cdot - 1})$

Step 5

Use logarithm rules to move $x - 6$ out of the exponent.

$(x - 6) \log_{3} (3) + \log_{3} (3^{x \cdot - 1})$

Step 6

Logarithm base $3$ of $3$ is $1$ .

$(x - 6) \cdot 1 + \log_{3} (3^{x \cdot - 1})$

Step 7

Multiply $x - 6$ by $1$ .

$x - 6 + \log_{3} (3^{x \cdot - 1})$

Step 8

Simplify each term.

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Step 8.1

Use logarithm rules to move $x \cdot - 1$ out of the exponent.

$x - 6 + x \cdot - 1 \log_{3} (3)$

Step 8.2

Move $- 1$ to the left of $x$ .

$x - 6 - 1 \cdot x \log_{3} (3)$

Step 8.3

Rewrite $- 1 x$ as $- x$ .

$x - 6 - x \log_{3} (3)$

Step 8.4

Logarithm base $3$ of $3$ is $1$ .

$x - 6 - x \cdot 1$

Step 8.5

Multiply $- 1$ by $1$ .

$x - 6 - x$

Step 9

Combine the opposite terms in $x - 6 - x$ .

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Step 9.1

Subtract $x$ from $x$ .

$0 - 6$

Step 9.2

Subtract $6$ from $0$ .

$- 6$